Lie models of simplicial sets and representability of the Quillen functor

Extending the model of the interval, we explicitly define for each n >= 0 a free complete differential graded Lie algebra L-n generated by the simplices of Delta(n), with desuspended degrees, in which the vertices are Maurer-Cartan elements and the differential extends the simplicial chain complex of the standard n-simplex. The family {L.}(n >= 0) is endowed with a cosimplicial differential graded Lie algebra structure which we use to construct two adjoint functors SimpSet reversible arrow(L)(<.>) DGL given by < L >. = DGL(L.,L) and L(K) = lim(-> K) L. This new tool lets us extend the Quillen rational homotopy theory approach to any simplicial set K whose path components are not necessarily simply connected. We prove that L(K) contains a model of each component of K. When K is a 1-connected finite simplicial complex, the Quillen model of K can be extracted from L(K). When K is connected then, for a perturbed differential partial derivative(a), H-0(L(K), partial derivative(a)) is the Malcev Lie completion of pi(1)(K). Analogous results are obtained for the realization < L > of any complete DGL.